Scattering States in AdS/CFT

TL;DR

This work constructs a precise bridge between AdS/CFT and flat-space scattering by identifying DeltaT-regulated multi-trace primary states in large-N CFTs that serve as in/out states in the AdS flat-space limit. Scattering amplitudes are extracted from smeared CFT correlators, with normalization anchored by disconnected correlators, and the flat-space S-matrix emerges as R->infinity and ER->infinity are taken in a controlled EFT regime. IR issues are tamed by AdS curvature, which acts as a universal regulator, and explicit checks in AdS3 demonstrate recovery of bulk T(s,t) for massless massless-contact interactions. The framework connects to broader program with Mellin amplitudes and suggests rich future directions, including nonperturbative S-matrix aspects and black-hole intermediate states accessible via CFT data.

Abstract

We show that suitably regulated multi-trace primary states in large N CFTs behave like `in' and `out' scattering states in the flat-space limit of AdS. Their transition matrix elements approach the exact scattering amplitudes for the bulk theory, providing a natural CFT definition of the flat space S-Matrix. We study corrections resulting from the AdS curvature and particle propagation far from the center of AdS, and show that AdS simply provides an IR regulator that disappears in the flat space limit.

Figures (3)

• Figure 1: The wavefunction $\langle 0| \phi(t,x) | \psi_{\omega, \tau} \rangle$ at $t=0$ corresponding to the state $| \psi_{\omega, \tau} \rangle$ constructed in eq. (\ref{['eq:smeared']}) by smearing the single-trace operator ${\cal O}(t,\widehat{x})$ acting on the vacuum. The choice of energy is $\omega = 100$ and of smearing window size is $\tau = 0.1$, in units of the AdS radius. The state is localized near the boundary, though the wavefunction falls to zero at $\rho=\pi/2$, as is necessary for a normalizable mode.
• Figure 2: Left: The bulk wavefunction $\langle 0 | \phi(x) | \psi\rangle$ at $t=0$, of a non-relativistic particle state $|\psi\rangle$ with a gaussian wavepacket, near the center of AdS$_3$. Right: The corresponding wavefunction $\langle0| {\cal O}(x) | \psi \rangle$ in the boundary theory, plotted along the surface of the boundary cylinder. Time runs upward along the cylinder, and the magnitude $|\langle 0 |{\cal O}(x) | \psi\rangle|$ of the wavefunction is its extent outward from the cylinder surface. Knowledge of the bulk wavefunction $\phi(x)$ and $\dot{\phi}(x)$ everywhere in AdS at a given time is enough to determine the state; by contrast, one needs the boundary wavefunction ${\cal O}(x)$ at all times in order to extract the same information.
• Figure 3: In this figure we have a flattened version of the AdS cylinder. The in and out states are prepared by integrating CFT operators over the grey regions with the smearing functions discussed in section \ref{['sec:States']}. Particles in AdS take a time $\pi R$ to travel from their closest approach to the boundary through the center of AdS and back out to the boundary, so we separate the in and out states by this time interval. We take $\Delta T \ll R$ so that the particles are created near the boundary, where they do not interact, and $\Delta T$ much larger than their wavelengths, so that their energies and momenta will be sharply defined in the flat space limit.